B6. Motion on precribed trajectory
b) Rigid body, which can rotate about a fixed axis, in motion
Duration of rotation. Angular velocity. Measure of magnitude of angle. Trajectory velocity
A body, which can rotate about a fixed axis, remains at rest only when the resultant of the forces acting on it intersects the axis. If it does not, the body rotates.
At the beginning of a rotation, all its points start to move simultaneously along circles, at the end, they stop simultaneously. Even if only one point of the body has completed its circle once, the body regains the same position as it had at the start. The time interval T between the start and the end of a full rotation is called its period. The greater the distance of a point from the axis, the longer its circular trajectory during T, whence it also has a larger velocity. Since the body is rigid, the paths and velocities of points at different distances from the axis are related. The body's rigidity forces all points which, for example, lie duringrest on a straight line, to do so also during rotation.
If during rotation about the axis (Fig. 87) which is perpendicular to the plane of the drawing at c, the point f moves from f1 to f 2, and e, d and all other points which lie on the same line as f, perpendicular to the axis, simultaneously move to e2, d2, ..., then e2, d2, ... lie on the same line through f2 and perpendicular to the axis.
In other words: for all points of the same line, the angle described by their perpendicular distances from the axis f1 , e1c, d1c, ··· during this time is the same. However, because the body is rigid, the same angle is also described by the perpendicular distances from the axis of rotation by all other points, that is, the entire body has turned during this time interval by this angle. The velocity at which it does this, that is, the ratio of the size of the angle over the time used is called angular velocity. We will return later on to this topic.
The magnitude of an angle (Fig. 87) measures the circular arc between its legs in its ratio to the length of the corresponding radius: In this case, the ratio of f1f2 to f1c or c1c2 to c1c. Since f1f2/f1c = c1c2/c1c = ···, this ratio is unique for that angle, whence it can be used as measure of the magnitude of the angle. - The length of the arc, described by the radius of 1 cm about c is in the same ratio as f1f2 to f1c, etc. Denoting this arc length by j, then f1f2/f1c = ··· = j /1 = j, that is, the magnitude of the angle is measured by the length of the circular arc, described with radius 1 cm about the vertex of the angle. The magnitude of the angle of 360º is the entire circumference of the circle with radius 1 cm. Hence
The length of the circular arcs f1f2, e1e2, etc. related to this arcj are f1f2 = f1c·j, e1e2=e1c·j, etc.
The path described by a point mrat distance r from the axis of rotation is r times as long as the arc j , which a point m1 describes at the distance 1 cm from the axis during the same time interval, whence mr has r times the velocity of m1. The velocity of m1 is measured by the length of the arc j, covered by it, in ratio to the time used. However, this arc is also the measure of the angle w, by which the body has rotated during the same time interval. The ratio of the length of the circular arc, covered by a point at distance 1 cm from the axis, to the time used is therefore simultaneously a measure for the velocity by which the entire body describes this angle. It is called its angular velocity.
If a body has undergone a complete turn, it has described the angle of 2p. If the duration of this turn is T sec, is has described in 1 sec the angle 2p/T. However, it describes in 1 sec the angle w - according to the definition of its angular velocity - whence w = 2p/T. If it undergoes in 1 sec n cycles, so that T = 1/n, then w = 2pn, where. n is the number of revolution per sec.
The discussion of the uniformity of velocity and acceleration explains the meaning of uniformity of angular velocity and acceleration. The same considerations which apply to motion along straight lines are also valid for circular arcs. If the point describes always at 1 cm from the axis of rotation at uniform angular velocity during unit time the arc of length w, then w is the angular velocity of the body. You must distinguish between angular velocity and path velocity. The path velocity is the ratio of the length of the arc rj covered during the time t used. The angular velocity of any point is the same as for any other point, that is, for the entire body. In contrast, the path velocity increases with r, that is, the further away a point is from the axis, the greater its velocity. If the angular velocity of every point is known, then also the path velocity of every point is known: It is equal to the product of w and its distance from the axis of rotation.
We are especially interested in points on a sphere rotating about a diameter, because we live on such a surface - at least we can conceive Earth to be such. Earth rotates by 360º in one day = 86400 sec (that is, half as fast as the hour-hand of a clock), its angular velocity is therefore 360º/86400. that is, only ¼ angle minute/sec - very small indeed (in arc measure: 2p/86400 = 0.000073). The path velocity of a point on its surface depends on its geographical latitude . It is w ·rcosj, where at the equator r = 6 378 388 m; it is therefore at the equator 73·10-6·6 378 388 465 m, at latitude 50º 299m. Points at the same latitude have the same path velocity; it changes with Latitude.
Coriolis acceleration (1835)
You do not note directly that the path velocities of points at the surface of rotating bodies, which lie at different distances from the axis of rotation, differ [at least not with those of a rigid body, the non-rigid body deforms - a noticeable action of this distinction]. However, you can note it indirectly, as a mobile mass - say a mass point - moves on a rotating body from one latitude to another. As a consequence of the rotation of the body, the mass point follows then relative to the points of the surface of the body a path which differs from that which it would pursue if the body did not rotate. The difference between the path velocities of points on the surface at different latitudes manifests itself.
This phenomenon is as well of interest to us, because we live on the surface of a rotating sphere and the water particles in rivers, the air particles in the atmosphere as well as bullets of long distance guns, etc. move from one circle of latitude to another; also the rotation of the plane of oscillation of a pendulum is here of concern as well as the deflection from the vertical of a freely falling body. On Earth's surface, you note the deviations only when the motion of mass points lasts a long time or when the velocities are very large, because, as we will see, the product of the velocity of the mass point and the angular velocity of the rotating body contribute here, but the angular velocity of Earth is very small.
However, on the surface of a body which we can rotate sufficiently fast, the deflections become noticeable quickly enough. - Following its discoverer Coriolis, the deflection motion - an acceleration - is called Coriolis acceleration and the force, by the action of which it is sensed, Coriolis Force.
A readily understood special case demonstrates the generation of the deviation (Fig. 88), A disk rotates in the direction of the arrow in the plane of the figure about the vertical through its centre M and on it moves a mass point starting from M. This point moves frictionless over the disk, that is, it follows only its inertia. Therefore we imagine it to move just above the disk parallel to it along a radial straight line at rest like on a rail. It moves at a constant velocity c starting at M.
At the end of the time interval t1, it reaches that point of the rail underneath which itwas located initially at the point B of the disk; however, during t1, the disk has turned away by the angle a below the material point so that the point of the disk, which initially was at B, is at B1.
At the end of the next time interval t2 = t1, the material point reaches on the rail that point, underneath which the point C of the disk was located initially; during the time interval t2 the disk has turned by the same angle a, whence the point of the disk, initially at C, is at C2 , etc. (The rotating disk and the rail at rest behave with respect to each other like the dial-plate and the hand of a horizontally placed watch, except that the dial-plate rotates and the hand rests).
The sidewards deviation of a point of the disk from the line, along which moves the material point, for example, the deflection CC2 of the disk point C is given by the product of the angle CMC2, measured as an arc, and the segment MC. The angle CMC2 is w t and the segment has the length ct, the deviation CC2 is therefore s = cw t = (2cw /2)t2. Obviously, this is a distance which has been covered with the acceleration 2cw during the time t (just imagine that you have placed next to it a distance fallen at the acceleration g!). The points of the disk deviate in the direction of the rotation of the disk with a velocity which is perpendicular to the direction of the velocity c of the material point and has the acceleration 2cw.
Discussion of Fig. 89: About the Coriolis deviation. (Coriolis) In the plane of the drawing rotates with constant angular velocity w a horizontally oriented disk about the vertical through M. From M a missile is launched radially at the target x resting on the disk with constant velocity c. The rotation of the disk deflects the points of the disk, over which the missile passes and the target towards the left side, so that the missile hits the disk to the right of the target. An observer who rests on the disk, that is an observer who partakes in the motion of the disk and does not sense it, draws the conclusion: "A force has diverted the missile towards the right hand side." This apparently present deflecting force is the Coriolis force . - The solid curve is the trajectory of the missile relative to the disk, which is assumed to be at rest, that is, for an observer, travelling with the disk, the apparent path of the missile; the broken curve, displays the actually displaced points of the disk due to the rotation during the motion of the missile.
This is how an observer on a immobile line imagines the movement of the material point. It appears quite differently when seen from the disk by an observer resting on it, that is, who takes part in the motion of the disk, but does not sense it! (Note the last point; just imagine: We take part in the rotation of Earth without noting it!) In order to get insight into what this observer sees, imagine a missile, just above a plane and flying parallel to it from the centre along a straight trajectory, the plane is infinite in all directions, nowhere are there points of reference, relative to which you can sense the rotation of the plane. The process will then appear to an observer who rests on this plane as discussed with Fig. 89 above. We ourselves are in just that position (at the centre of the horizontal plane, which rotates about us.)
We are mainly interested in the Coriolis deviation because we live on the surface of rotating Earth and can explain by means of it certain changes in direction of geophysical processes on its surface. A mass which moves along the surface of Earth (air in winds, water in rivers) does not move without friction and therefore the centrifugal force also contributes. But these two phenomena are so little active that we may neglect friction as we have initially assumed. But we must note one point: At a point of Earth's surface at latitude j, we must only take into account the angular velocity's component w ·sinj .Among the geophysical processes in which the Coriolis Deviation has a role is first of all the deflection of wind as a result of Earth's rotation.
The steady difference between the heat transfer at higher and lower latitudes generates a basic circulation in the atmosphere. The excess heat received by the equatorial region induces a circulation: In the upper layers of the atmosphere, the winds move polewards from the Equator, they move simultaneously in the lower layers from the Poles to the Equator. But Earth's rotation complicates the process. The upper, polewards moving air approaches gradually the axis of rotation, increases thereby in the upper latitudes its velocity relative to points on Earth's surface, eventually moves ahead of them and becomes a West wind. For a similar reason becomes the lower, to the Equator flowing air (because it remains behind the points on Earth's surface) in higher latitudes an East wind. That was in 1930 (since Ferrel 1860) the basic concept of the cause of the deflection of wind due to Earth's rotation.
Baer's law (1860)
Baer has explained peculiarities of the formation of the shores of meridi0nally running rivers in Russia by the Coriolis Deviation (Baer's Law): He found that in Siberia the right shores of many rivers in lowlands were for long distances steep and high, the left banks low and flat. He explained this phenomenon by the increased pressure of the current towards the shore due to Earth's rotation; it forces the river to shift its shore towards the right side as long as no higher land opposes it
The flowing water, which moves from the South to North brings with it higher velocities than prevail in Northern regions of Earth and thus presses against the Eastern shores, because Earth rotates towards the East, and with it the excess water which comes from the lower to the higher latitudes. Conversely, water flowing from the polar regions towards the Equator will arrive there at lower speeds and press against the Western banks. However, on the Northern hemisphere, the Eastern shore of rivers flowing North is on the right and those of river flowing South on the left. Hence on the Northern hemisphere are the Eastern shores of rivers, which flow more or less meridionally, the affected, steeper and higher ones, the Western shores low and flooded, and indeed more so as the directions of rivers approach Meridians, so that along rivers and river sections which are almost totally meridional other disturbing effects show off less. - If this explanation is correct, the situation on the Southern must be reverse. Von Baer has shown that this is the case.
Also rotation is visualized by a directed segment. It is placed along the axis of rotation, its length equals the angular velocity w and it is given an arrow in the direction which forms with the sense of turn a right- hand screw (Fig. 90).
The rotor is an axial vector (in contrast to a polar vector). It is unimportant, from which point O of the axis the vector is drawn. - Moreover, the connection of the path velocity of a point with the angular velocity of a body can be envisaged. A point P which is at distance r1 from the axis and at the instant is on the ray r from O to P and makes the angle a with the vector w, has the path velocity v = wr1= wrsina. On the right hand side, you have twice the area of the triangle, formed by w and r. It is called the vector product and represented by the directed segment v.
You make the magnitude of v equal to twice the area of the triangle and place it perpendicular to the plane of the triangle in such a manner that the arrow and a form a right hand screw , which moves the vector w along the shortest path into the direction of the vector r.
At times, a body may rotate simultaneously about several axes. This vector representation allows its rotation about several axes to be composed into one about a resultant axis. What is the meaning of simultaneous rotation about several axes?
You should think here of the top with which you played as a child, the tip of which remains in place while it rotates (Fig. 91): It rotates 1. about its axis of symmetry (axis of figure) and simultaneously 2. this axis rotates about the vertical axis, passing through its point of support, and describes a cone (the point of the top is the tip, the vertical the axis of the cone.) As the top slows down, its axis swings in addition towards the vertical and back, that is, 3. it turns about a horizontal line through its support point. In other words: Every mass point of the body - only not the point of support - rotates simultaneously about several axes. All these rotations combine into the motion of the staggering top which struggles with falling over. The axis of symmetry of the top is no longer the preferred axis of rotation of the system, it is only one of several.
How to combine rotations? How to find the resultant rotation and its axis? We will discuss here only the fundamental rule according to which two rotations about two axes which pass through the same point are combined into a resultant rotation (and how a rotation is decomposed into two components). Let u1 and u2 be the two vectors of two rotations. Each determines the axis and angular velocity of corresponding rotations: Each of these rotations gives every mass point of the body a definite velocity. The composition of the rotations tells you the velocities which the mass points of the body have, generated by these two rotations. 1. Also the resultant motion is a rotation of the body about an axis; 2. The angular velocity and direction of the axis of the resultant rotation is given by the rotor which is the result of addition of the vectors u1 and u2 according to the parallelogram law.
By the same law, you can decompose a rotation about an axis into several simultaneous rotations (about several axes). An example follows: It may prove useful to conceive the rotation of Earth as having beern decomposed into two components, in fact, whenever you want to compute processes, the cause of which is the rotation of Earth's axis and the magnitude of which depends on the Latitude of the site of observation. Examples are Foucault's pendulum experiment, Hagen's experiment to prove the rotation of Earth by the isotomeograph, etc.
You represent Earth's angular velocity w (Fig. 92) by the vector v which lies along the axis of Earth and points northwards from Earth's centre; its length corresponds to the angle 2p/86400, covered during one second. ( In the numerator is the angle covered during a mean Sun Day, measured on a circle with radius 1, in the denominator the number of seconds per day). You now decompose the vector v in the accustomed manner into two components c1 and c2. The first component is a rotation of the horizontal plane (azimuthal rotation) at the point of observation A at latitude j about the vertical there; it has the angular velocity w1 = w sinj, it only vanishes at the equator (j = 0) and becomes at the poles Earth's complete rotation. The second component (vertical rotation) turns the horizon at the angular rotation w 2 = w cosj about an axis, which passes through Earth's centre parallel to the direction North at the point of observation; it only vanishes at the two poles and represents at the Equator the complete rotation.
For example, this decomposition of Earth's rotation shows why Foucault's pendulum and the Isotomeograph demonstrate best at the poles the turning of Earth's axis, and not at all at the Equator.
Centripetal and centrifugal force
If the body turns due to the action of a force, say, like the hand of a clock, as the force stops to act, it continues to turn in the same direction and, indeed, with that angular velocity which it had as the force stops.Therefore rotation with a uniform angular velocity is an inertial motion: The body maintains its state of motion in direction and angular velocity, unless an external force stops it (a possibility which will be excluded here). But this quasi-inertial motion differs from the real inertial motion: The individual mass points move with uniform velocity, but in circles. However, a point can leave a straight line only if a force acts. Hence we must assume that a force acts and indeed deflects it radially towards the centre of the circle which it describes from that straight line which it would have followed had it been able to act inertially - this is the centripetal force.
The force resists inertially the action of this deflecting force, diametrically opposite to the deflecting force: This resistance attempts to inhibit an approach of the point towards the centre of the circle; it acts radially, but away from the centre, whence it is called the centrifugal force. This centrifugal force has therefore only significance relative to the simultaneously acting centripetal force. It is the reaction, if you consider the centripetal force to be the action; as always in the case of action and reaction, it occurs concurrently with the centripetal force, that is, it must appear and vanish simultaneously with it and also for the same reason be equally large.
In order to understand how the centripetal force acts, you must understand the relationship of a mass point to the centre of its trajectory. Fig. 93 shows a plane at right angle to the axis of the body; C is its intersection with the axis, the circle is the path of a point and the curved arrow the direction of the rotation. The mass point, due to its inertia, wants to continue along the tangent, the centre, as a point of the fixed axis, must stay in place. Both belong to the same rigid body, whence they cannot change the distance between them. The centre is forced to stay in place and not change its distance to the mass point, the mass point not to change its distance from C, but nevertheless to advance. The circulating mass point therefore tends to displace the centre (maintaining its distance to it).
Since you can imagine its point of attack shifted anywhere along its direction, the centrifugal force also tends to increase the distance between the mass points and the centre. It thus competes with the forces of rigidity of the body and its cohesion. If the body were perfectly rigid, this property would make the centrifugal force ineffective, however, no body is totally rigid. The mass points move therefore more or less under the action of a sufficiently large centrifugal force, that is, the body gives in, deforms and can even split up. You can envisage those forces which are essential for the rotation of a rigid body (centripetal and centrifugal forces, action on the centre, competition between cohesive forces) by a meaningful process, which however disregards rigidity, but realizes the essential aspects of rotation.
If you swing by hand a heavy body on a rope in a circle, the rope becomes stretched as a result of the centrifugal force and the hand senses a pull towards the body. It must pull it strongly inwards (centripetrally), in order not to be oulled outwards (centrifugally). If you swing the body faster and faster, you increase the centrifugal force. The pull, felt by the hand, becomes larger, the rope tighter and eventually it breaks, the motion in a circle and the pull on your hand end simultaneously and the body flies off.
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